Nanoscale Dispensing of

Single Ultrasmall Droplets

INAUGURALDISSERTATION

zur

Erlangung der Wrde eines Doktors der Philosophie

vorgelegt der

Philosophisch-Naturwissenschaftlichen Fakultt

der Universitt Basel

von

Andr Meister aus Matzendorf (So)

Basel, 2005

Genehmigt von der Philosophisch-Naturwissenschaftlichen Fakultt auf Antrag

der Herren

Professor Dr. E. Meyer

Privatdozent Dr. H. Heinzelmann

Basel, den 21. September 2004

Prof. Dr. Hans-Jakob Wirz, Dekan

i

Abstract Nanoscale dispensing (NADIS) is a novel technique to deposit material at micrometric and

submicrometric dimensions. It has great flexibility in feature shape and choice of deposited material. Due to its expected low cost and short turn-around time, it has the potential to be an interesting tool complementary to standard lithographic processes. Furthermore, NADIS has a great potential in the creation of high-density microarrays used in proteomics or genomics.

The key feature of NADIS is the deposition of a liquid through an aperture created in a scanning force microscopy probe tip. The liquid is loaded into the hollow back of the pyramidal probe tip. Upon contact, liquid is transferred from the tip to the substrate surface. The transfer of liquid occurs without any external pressure. The control of the NADIS probe displacement is achieved using a standard atomic force microscope.

Two different approaches to fabricate tips with apertures were investigated. The first approach relies on opening the tip during the microfabrication of the probe, whereas the second approach implies a modification of commercially available probes by focused ion beam milling. Both kinds of probes have shown their ability to perform successfully the dispensing of droplets.

Nanoscale dispensing has been demonstrated for deposition of ultrasmall single droplets with volumes down to 5 attoliters in a controlled way and with high lateral accuracy. The smallest droplet spacing that has been achieved was less than 500 nm. The size of the droplets and the possible droplet density are largely dependent on the aperture diameter and on the surface wettability.

Nanoparticles and fluorescent molecules were also dispensed. In such cases, the liquid is used as a transport medium for the substances to be deposited.

By moving the NADIS probe during contact on the substrate, it was possible to write features such as lines with sizes that can be as small as 400 nm.

Some theoretical aspects are discussed, in particularly the capillary forces associated with axisymmetric liquid menisci. Experimentally measured capillary forces during the dispensing are compared with theoretically determined values.

ii

Acknowledgments Many people have helped me, directly or indirectly, to bring to end my thesis presented herein. I would like to express my sincere gratitude to my supervisor, PD. Dr. Harry Heinzelmann from

CSEM SA, who gave me the opportunity to do my graduate study in the best conditions. He offered me the possibility to work in an independent way, and was always available to answer my question, despite his busy agenda.

I would like to address my cordial thanks to Prof. Dr. Ernst Meyer from the University of Basel,

who spontaneously accepted to be my Faculty Responsible. I would also like to express my thanks to Dr. Terunobu Akiyama, Prof. Dr. Urs Staufer and Prof.

Dr. Nico De Rooij, from IMT at the University of Neuchtel for supplying me with the probes with microfabricated apertures and for valuable discussions, and to Phillippe Gasser from EMPA for operating the FIB instrument.

I am particularly indebted to the entire nano and bio group, especially to Sivashankar

Krishnamoorthy and to Dr. Christian Hinderling for providing me with the substrates made of PS-b-P2VP micelles, to Dr. Raphal Pugin, my superior, and to Dr. Martha Liley for their many enlightening discussions and continuous encouragements, but also to Dr. Caterina Minelli, Dr. Kaspar Cottier, Dr. Rolf Eckert, Dr. Rino E. Kunz, Dr Guy Voirin, Myriam Losson, Vronique Monnier, Eric Bernard, Nicolas Blondiaux, and Ral Ischer. All have participated to continuously maintain a friendly environment.

I am very grateful to Prof. Dr. Jrgen Brugger, Prof. Dr. Peter Vettiger and Prof. Dr. Rolf P. Steiger

for their numerous interesting discussions. A special thanks to Top Nano 21, to the National Center of Competence in Research (NCCR) in

nanoscale science, and to the Swiss Federal Office for Education and Science (OFES) in the framework of the EC-funded project NaPa (Contract no.NMP4-CT-2003-500120) for their partial financial support.

And last but not least, I would like to sincerely thank my wife Carole, who had always trust in my

work, even in the periods where I was full of doubt. She gave me continuous support, and was of valuable assistance.

iii

Table of contents

1. Introduction ..................................................................................................... 1 2. Theoretical background.................................................................................. 5

2.1. Forces that hold together atoms or molecules in a condensate.......................................... 5 2.1.1. The Origins of the forces ............................................................................................... 5

2.1.1.1. Electrostatic and electrodynamic interaction ......................................................... 6 2.1.1.1a. Purely electrostatic interactions ...................................................................... 6 2.1.1.1b. Polarization forces .......................................................................................... 7 2.1.1.1c. London dispersion forces................................................................................ 7

2.1.1.2. Van der Waals forces ............................................................................................. 8 2.1.2. Molecular and atomic condensates ................................................................................ 9

2.1.2.1. Intramolecular interactions (atoms form molecules) ............................................. 9 2.1.2.2. Intermolecular interactions (molecules form a condensate) ................................ 10 2.1.2.3. Interatomic interactions (atoms form a condensate) ............................................ 10

2.2. Interactions between macroscopic bodies........................................................................... 12 2.2.1. Surface Free Energy and Cohesion.............................................................................. 12 2.2.2. Interfacial Energy and Adhesion.................................................................................. 14

2.2.2.1. Film pressure........................................................................................................ 15 2.2.3. Forces between condensed bodies ............................................................................... 16

2.2.3.1. Adhesion forces between macroscopic objects.................................................... 16 2.2.3.2. Van der Waals forces between macroscopic objects ........................................... 16

2.3. Liquids on surfaces ............................................................................................................... 18 2.3.1. Laplace pressure........................................................................................................... 18 2.3.2. Contact angle ............................................................................................................... 20

2.3.2.1. Spreading coefficient ........................................................................................... 22 2.3.2.2. Influence of the surface roughness on the contact angle ..................................... 23

2.3.2.2a. Wenzel's approach ........................................................................................ 23 2.3.2.2b. Cassie's approach.......................................................................................... 23

2.3.2.3. Contact line tension ............................................................................................. 24 2.3.2.4. Contact angle hysteresis....................................................................................... 25

2.3.3. Hydrostatic equilibrium ............................................................................................... 26 2.3.4. Numerical construction of an axisymmetric liquid-gas interface ................................ 27

2.3.4.1. Shapes of pendent and sessile drops .................................................................... 28 2.3.4.2. Axisymmetric shapes when neglecting gravity ................................................... 29

2.3.4.2a. Axisymmetric shapes with a zero mean surface curvature ........................... 30 2.3.4.2b. Axisymmetric shapes with a positive mean surface curvature ..................... 30 2.3.4.2c. Axisymmetric shapes with a negative mean surface curvature .................... 31

2.3.4.3. Meniscus stability ................................................................................................ 32 2.3.5. Capillary forces............................................................................................................ 33

2.3.5.1. Capillary force due to the surface tension............................................................ 33 2.3.5.2. Capillary force due to the Laplace pressure......................................................... 34 2.3.5.3. Capillary force for axisymmetric meniscus ......................................................... 34

2.3.6. Thermodynamic equilibrium and spontaneous condensation ...................................... 35 2.3.6.1. Capillary force induced by a spontaneously condensed meniscus....................... 37

iv

3. Experimental setup ....................................................................................... 38 3.1. Atomic force microscope...................................................................................................... 38

3.1.1. Typical AFM set-up..................................................................................................... 38 3.1.1.1. AFM operating modes ......................................................................................... 39 3.1.1.2. Hysteresis of the piezoelectric scanner ................................................................ 40

3.2. Contact angle measurement................................................................................................. 41 3.2.1. Surface tension measurement of liquids ...................................................................... 42

3.3. Miscellaneous ........................................................................................................................ 43 4. Probes realization.......................................................................................... 44

4.1. Microfabrication................................................................................................................... 44 4.2. Probes with microfabricated apertures .............................................................................. 46

4.2.1. 1st version of microfabricated probes........................................................................... 46 4.2.1.1. Cantilever stiffness and loading area ................................................................... 47

4.2.2. 2nd version of microfabricated probes .......................................................................... 48 4.3. Probes with apertures opened by FIB................................................................................. 50

4.3.1. Reflection of the laser beam for V-shaped cantilever with loading area ..................... 55 5. Methods .......................................................................................................... 57

5.1. List of chemicals.................................................................................................................... 57 5.1.1. Choice of liquid to be deposited .................................................................................. 57 5.1.2. List of solvents and chemicals ..................................................................................... 57

5.2. Preparation of the substrate ................................................................................................ 59 5.2.1. Substrate materials ....................................................................................................... 59 5.2.2. Substrate cleaning ........................................................................................................ 59

5.2.2.1. Piranha ................................................................................................................. 59 5.2.2.2. SC-1 ..................................................................................................................... 60 5.2.2.3. Oxygen plasma .................................................................................................... 60

5.2.3. Surface treatment ......................................................................................................... 60 5.2.4. Behavior of the deposited liquids on the substrates ..................................................... 61

5.3. Probe preparation and characterization ............................................................................ 63 5.3.1. Preparation of the loading area for the FIB-modified probes ...................................... 63

5.3.1.1. Hydrophobic-hydrophilic contrast of gold relative to silicon nitride................... 63 5.3.2. Spring constant determination ..................................................................................... 64

5.3.2.1. Resonance frequency ........................................................................................... 65 5.3.2.2. Thermally induced oscillation.............................................................................. 69

5.4. Dispensing of the droplets .................................................................................................... 73 5.4.1. Loading of the probe.................................................................................................... 73

5.4.1.1. Shape of the loading area..................................................................................... 73 5.4.1.2. Filling of the hollow tip ....................................................................................... 74 5.4.1.3. Determination of the amount of loaded liquid. .................................................... 75

5.4.1.3a. Evaporation of the liquid from the loading area ........................................... 76 5.4.2. Transfer of liquid characterization by deflection-distance curves ............................... 77 5.4.3. Measurement of the capillary force ............................................................................. 79 5.4.4. Dispensing of the droplets to form a pattern................................................................ 79 5.4.5. Cleaning of the probe................................................................................................... 80

5.5. Characterization of the droplets.......................................................................................... 81

v

5.5.1. How to determine the volume of the deposited drop ................................................... 81 5.5.1.1. By the droplet dimension..................................................................................... 81 5.5.1.2. By the shift in resonance frequency..................................................................... 82 5.5.1.3. By the capillary force........................................................................................... 82

5.5.2. Visualization of the deposited droplets ........................................................................ 82 5.5.2.1. Optical imaging.................................................................................................... 82 5.5.2.2. AFM imaging....................................................................................................... 82

5.5.2.2a. Localization of the deposition area ............................................................... 83 5.5.2.2b. What is seen after the deposition .................................................................. 83

6. Results and discussion................................................................................... 84 6.1. Typical results of droplet deposition................................................................................... 84

6.1.1. Deposition using probes with microfabricated aperture .............................................. 84 6.1.2. Deposition using FIB-modified probes........................................................................ 85

6.2. Parameters that influence the amount of deposited liquid ............................................... 86 6.2.1. Aperture size and tip geometry .................................................................................... 86

6.2.1.1. Position of the aperture in the FIB-modified probes............................................ 87 6.2.2. Surface energy of the sample ....................................................................................... 88

6.2.2.1. Hydrophilic substrate with hydrophilic outer tip wall ......................................... 89 6.2.3. Contact time and number of contacts........................................................................... 91

6.2.3.1. Influence on the droplet diameter ........................................................................ 91 6.2.3.2. Influence on the droplet height ............................................................................ 92

6.2.4. Withdrawal speed......................................................................................................... 94 6.2.5. Laplace pressure........................................................................................................... 95

6.3. Capillary force measurements and meniscus shapes......................................................... 96 6.3.1. Example of a meniscus shape with a probe having a microfabricated aperture .......... 98 6.3.2. Example of a meniscus shape with a FIB-modified probe .......................................... 99

6.4. Evaporation of the droplets and residual dots ................................................................. 100 6.4.1. Evaporation time........................................................................................................ 100 6.4.2. Residual dots after the evaporation............................................................................ 104

6.4.2.1. Position of the residual dot ................................................................................ 107 6.5. Deposition of molecules, and nanoparticles...................................................................... 109

6.5.1. Deposition of droplets with fluorescent molecules.................................................... 109 6.5.2. Deposition of droplets with nanoparticles ................................................................. 109

6.5.2.1. Deposition of droplets with 365 nm wide silicon dioxide particles................... 110 6.5.2.2. Deposition of droplets with 20 nm wide polystyrene nanoparticles .................. 111

6.5.2.2a. Dispensing with probes having a microfabricated aperture........................ 111 6.5.2.2b. Dispensing with probe modified by FIB..................................................... 112

6.6. Deposition on hydrophobic substrates for small droplet spacing .................................. 115 6.7. Modification of block-copolymer micelle layers .............................................................. 117

7. Conclusion.................................................................................................... 121 7.1. Application of nanodispensing........................................................................................... 122 7.2. Outlook ................................................................................................................................ 123

8. Annexes......................................................................................................... 124 8.1. Annex 1 ................................................................................................................................ 124

8.1.1. Youngs equation ....................................................................................................... 124

vi

8.2. Annex 2 ................................................................................................................................ 126 8.2.1. Program...................................................................................................................... 127

8.3. Annex 3 ................................................................................................................................ 130 8.4. Annex 4 ................................................................................................................................ 134

8.4.1. Evaporation of a sessile droplet ................................................................................. 134 8.4.1.1. With a pinned three-phase contact line.............................................................. 134 8.4.1.2. With a constant contact angle ............................................................................ 134

9. References .................................................................................................... 136

1

1. Introduction

Liquid handling has various aspects and is involved in a large range of activities, from water distribution in a building to the precise deposition of a drop of ink onto paper in an inkjet printer. The devices used for the different applications of liquid handling are based on diverse technologies as various as sanitary facilities or high-tech microelectromechanical systems (MEMS). The usual containers for liquid found in daily life permit the control of volumes that are between some liters and a centiliter. A controlled handling of smaller volumes of liquids, e.g. for the preparation of a solution with a precise concentration in a chemical laboratory, usually involves syringes or pipettes. Medical syringes handle volumes in the milliliter range and above, pipettes deal with volumes in the milliliter to microliter range, and the sub-microliter domain is accessible with Hamilton syringes [1].

The pharmaceutical industry commonly uses microtiter plates (or microplates) in combinatorial chemistry or in high throughput screening. Standard microplates have 96, 384 or 1536 wells, with volume of liquid in each well of about 100, 20 and 5 microliters respectively. Because manual pipetting is limited in turn-around time, the microplate platform uses dispensing robots with many pipettes in parallel for a higher throughput and automated use [2]. Normal pipetting systems are limited to volumes above 200 nanoliters, because of problems of drop release from the pipette nozzle.

Another dispensing method used with microplates is the dosing system. In this method, the liquid is pressurized using a pump unit, and a high-speed valve releases the liquid through a narrow orifice. The liquid is expelled as a free flying jet. The volume of the dispensed liquid is controled through the opening time of the valve. Such dispenser robots are able to control volumes down to 25 nanoliters [3].

The emergence of the microarray platform in the nineties has further increased the throughput of chemical and biological analyses, and, in parallel, considerably reduced the consumption of chemicals in the assays. Microarrays are highly parallel biological assays, which rely on the affinity between two molecules (receptor and analyte molecules). The two molecules can be complementary ssDNA (single strand deoxyribonucleic acid), or antigen antibody pair. The two complementary molecules react together with a lock-and-key effect. During the creation of microarrays, different receptor molecules are fixed on a substrate (usually a glass slide) to form a matrix of spots with different functionalities. The sample (liquid containing the analyte molecules) is brought into contact with the fixed receptor molecules. If two complementary molecules come close together, they bind and remain bound during rinsing and washing of the substrate. Detection is performed using e.g. fluorescent or chemiluminescent labels. The readout is done by the determination of the fluorescence or chemiluminescence pattern relative to the positions of the spots of receptor molecules.

Two main approaches are available for the production of microarrays. The first approach is to synthesize a DNA sequence in situ directly on the substrate [4]. The second approach is to deliver the receptor molecule to the substrate. The delivery of the receptor molecules can again be separated into two classes, namely non-contact and contact printing methods.

The most common non-contact printing tool is probably the head of an inkjet printer, which ejects droplets from a nozzle onto the substrate. Such droplets typically have volumes down to a few picoliters. Although the inkjet technique is mainly used for color printing, it is also involved in other domains, such as the fabrication of microarrays for biomedical and pharmaceutical applications. The

2

droplet is ejected from the head by applying a short pulse of pressure to the liquid located in a chamber that is connected to the nozzle by a small channel. The pressure in the chamber is built up either by a compression of the liquid volume by the action of a piezoelectric element, or by the expansion of a steam bubble produced by local heating. For the specific application of microarray printing, MEMS based drop-on-demand dispensers have also been developed, and allow the delivery of larger droplets, up to some nanoliters [5].

Contact based deposition of liquids can be realized by pin spotting or pin-and-ring spotting. These techniques use an array of pins. Each pin is dipped in a different solution containing the different receptor molecules. A mechanical contact between the wetted pins and the substrate takes place, and the liquid is transferred during the contact [6]. Spotting pins produce dots with typical sizes of about hundred microns.

These different methods are summarized in Figure 11.

Manual pipetting

Pin printer

1 100 101 100 101 100 10 1 100 10 11 100 10100 10milliliter microliter nanoliter picoliter femtoliter attoliter

Dispensing robots for microplates

Ink jet

1 mm 100 m 10 m 1 m

1 l 1 nl 1 pl 1 fl

(1 mm)3 = 1 microliter (100 m)3 = 1 nanoliter (10 m)3 = 1 picoliter (1 m)3 = 1 femtoliter (100 nm)3 = 1 attoliter

100 nm

1 al

Figure 11: Top: Overview of various dispensing methods with their respective ranges of volume handling. Bottom: In order to get a better feeling for the dimensions involved in the description of small volumes, consider a cube with an edge length of one millimeter, which corresponds to the volume of one microliter (l). If the length of the cube is decreased several times by a factor ten, one obtains the sequence shown. One can compare these volumes to objects found in nature and in daily life. E.g., a few human red blood cells fit into one picoliter (1 pl = 10-12 liter), a droplet expulsed from an aerosol spray has a volume of some tens of femtoliters and one attoliter (1 al = 10-18 liter) corresponds roughly to the volume of a large virus.

Using microfabrication techniques, news dispensers have been created that allow the dispensing of droplets in the picoliter and femtoliter range, with spot diameters of 2 to 3 m [7,8].

3

Dispensing methods based on nanopipettes have also been developed. In one approach, the nanopipette is filled with the material to be deposited. The material is then pushed out by applying a pressure at the larger opening of the nanopipette while the tapered end of the nanopipette is in vicinity of the substrate surface [9,10]. The liquid inside the nanopipette can also be delivered by a combination of electroosmotic flow, electrophoresis, and dielectrophoresis [11]. In such cases, the nanopipette must be immersed in a bath of ionic solution.

Driven by the motivation to push back the limit of liquids dispensing, the aim of this thesis was the

deposition of volumes of liquid in the attoliter range. By deposition of attoliter volumes, one means not only small volumes, but also a precise manipulation of the liquid. The goal was to dispense individual droplets on-demand, and to deposit them with a high lateral accuracy onto specific locations of a substrate, e.g. to form an array with a high density of droplets.

The interest was to investigate the feasibility of liquid handling with small volumes, to reach high spot densities and for an economical use of the liquid, and in parallel to understand the different parameters and limitations that are involved in it.

The method presented here and developed during this thesis, called nanoscale dispensing

(NADIS), is a contact-based dispensing of ultrasmall droplets. The probe used in nanoscale dispensing, the nanodispenser, is based on a similar concept to a probe for scanning near field optical microscopy, which has an optical aperture. The nanodispenser consists of a flexible cantilever which ends in a tip, similarly to an atomic force microscopy probe, but with an aperture at the tip. For the deposition process, the back of the tip is loaded with the liquid to be dispensed. When the tip comes into contact with the substrate, the liquid wets the substrate surface. As the probe is withdrawn, the meniscus of liquid which forms a bridge between the tip and the surface, is stretched and finally breaks, and a small droplet remains on the substrate. This transfer of liquid is based only on capillarity and on surface wettability. No external pressure is applied to the liquid. A schematic of the process is shown in Figure 12.

Figure 12: A conceptional sketch of nanoscale dispensing. The nanodispenser, a hollow AFM tip with an aperture, is loaded with a liquid. By bringing the tip briefly into contact with the substrate, some liquid is transferred to the surface.

4

The NADIS probe is mounted on a conventional atomic force microscope (AFM), and force feedback allows control of the tip-to-sample distance during the deposition process. The control software permits the addressing of individual points on the scanned surface where specific actions can be performed, such as lowering the probe to make contact with the substrate.

The NADIS method presented here permits the on-demand deposition of small individual droplets in an ambient environment at a predetermined location on a sample. A high lateral positioning accuracy is provided by the piezoelectric scanner of the AFM itself. Furthermore, a small droplet size combined with small droplet spacing allows the creation of arrays with high droplet densities.

Nanoscale dispensing is a maskless dispensing method, and offers a high degree of flexibility in the pattern design, and in the liquid to be used. The method can be extended to dispense other materials, such as biological molecules or nanoparticles. In such cases, the liquid acts as a transport medium for the components suspended in it.

A paper describing the method and the first results can be found in reference [12]. Although the concept is simple, a deeper understanding of the phenomena involved requires some

theoretical consideration. It is known that, in contradiction to the macroscopic world that is dominated by volume forces, in a microscopic world surface forces prevail. This is because, as dimensions decrease, the surface-to-volume ratio increases. Thus, the surface tension of liquids or the driving forces of capillarity are predominant when ultrasmall volumes of liquids are considered. The next chapter will address some theoretical aspects related to these phenomena.

5

2. Theoretical background

In this chapter, some physical aspects will be considered to get a better understanding of liquids, such as the shape of small droplets and their interaction with solid substrates, and capillary forces. Our understanding of liquids relies mostly on an analysis of surface and interfacial energies, and their implications. Interfacial energies originate in interatomic and intermolecular forces. Therefore, the chapter will start with a description of those forces.

It will be shown that the description of capillary phenomena is based on three fundamental equations. The first takes into account the hydrostatic equilibrium across the liquid-gas interface (Laplace pressure), the second deals with the thermodynamic equilibrium of the interfacial energies and leads to an understanding of the contact angle at the three-phase contact line (Young's equation), and the last treats the thermodynamic equilibrium over the liquid-gas interface with respect to evaporation or condensation (Kelvin's equation).

2.1. Forces that hold together atoms or molecules in a condensate

2.1.1. The Origins of the forces The various attractive forces that hold together molecules or atoms to form a liquid or solid

condensate, are different in nature. One can mention those arising from electrostatic and electrodynamic interactions, involving charges, permanent dipoles and induced dipoles. Ionic and van der Waals forces belong to this category, as well as those interactions responsible for hydrogen bonds. Another category of attractive forces is formed by the short-range forces that are quantum mechanical in nature, such as covalent and metallic bonds.

The repulsive forces that balance the attractive forces between atoms at very small interatomic distances are mainly due to the overlap of electron clouds, according to the Pauli exclusion principle. These interactions are also known as exchange, hard-core or steric repulsion. These forces have a very short range, and increase very sharply as two molecules or atoms are brought together.

At this point, to get a better overview, one should roughly classify the different interactions with the different actors involved, i.e. atoms and molecules:

- Intramolecular interactions are involved in the bonding between the atoms to form a molecule. - Intermolecular interactions are involved in the bonding between molecules to form a molecular

condensate. - Interatomic interactions are involved in the bonding between atoms to form an atomic

condensate. These different interactions are shown in Figure 21. One kind of interaction can be present in more that one of the three mentioned groups.

6

Atoms

Molecules- covalent (H2)- ionic (HF)- H-bond (DNA)

Atomic condensate- metallic (Fe)- covalent (C) - ionic (NaCl)

Molecular condensate- van der Waals (hydrocarbon)- H-bond (H2O)

Intertomicinteraction

Intermolecularinteraction

Intramolecularinteraction

Figure 21: Rough and non-exhaustive classification of the interactions involved in intramolecular, intermolecular and interatomic bonds.

Firstly, some of the different interactions will be described, without taking account on their organization in the three different groups.

2.1.1.1. Electrostatic and electrodynamic interaction

Electrostatic and electrodynamic interactions can be subdivided into the following subclasses:

2.1.1.1a. Purely electrostatic interactions Purely electrostatic interactions are those involving the charge of ions and the permanent dipole of

polar molecules. Charge-charge, charge-dipole and dipole-dipole interactions belong to this category. For dipoles, one must differentiate between angularly fixed and freely rotating dipoles. In general, a dipole with a moment, u, in an electric field, E, is angularly fixed when the thermal energy kBT is much weaker than the interaction energy uE, or freely rotating if uE kBT.

A special kind of electrostatic interaction is the Keesom interaction, also called the orientation interaction. This occurs when two freely rotating polar molecules mutually interact. The Keesom interaction energy is a Boltzmann-weighted and angle-averaged energy, and depends on the thermal energy kBT.

Another electrostatic interaction is the hydrogen-bond, which is a dipole-dipole interaction. This interaction exists between strongly electronegativea) atoms, such as O, N, F or Cl, and hydrogen atoms covalently bound to similar atoms. These bonds always involve hydrogen atoms, which can be easily positively polarized and are exceptionally small. The intramolecular covalent bond between H and an electronegative atom (e.g. O-H+) is highly polar. Additionally, due to the size of the electron depleted hydrogen atom, the dipole feels a very strong electrostatic field because it can get very close to a nearby negatively charged electronegative atom of another molecule, leading to a hydrogen-bond ( O-H+O-). Such hydrogen-bonds are highly directional. They are the origin of the particular properties of water,b) and play a predominant role in biology, e.g. binding two polynucleotide chains to form the inter-twisted DNA double helix (see Figure 22), or governing the folding of macromolecular structures like proteins.

a) The electronegativity of an atom is its affinity to capture additional electrons. b) Water has an unexpectedly high melting and boiling point, compared with its low molecular weight. H-bonds can also explain the density maximum at 4C, and the phenomena that the solid (ice) is lighter than the liquid.

7

1st strand

2nd strand

Figure 22: Two deoxyribonucleic acid (DNA) strands forming a helix are held together through H-bonds (taken from [13]).

2.1.1.1b. Polarization forces These forces arise when a dipole moment, uind, is induced in a molecule by the electric field E of a

nearby charge or a permanent dipole. The electronic polarizability, 0, of a molecule results from the displacement of the negatively charged electron clouds relative to the positively charged nucleus in the presence of an electric field, leading to an induced dipole moment uind = 0E. Both polar and non-polar molecules can be polarized in this way.

Such an interaction between the freely rotating dipole of a polar molecule and the induced dipole of a neighboring molecule is known as a Debye interaction.

Freely rotating polar molecules with a dipole moment, u, exposed to an electric field can have an additional orientation polarizability, ori. The time-averaged dipole moment of a freely rotating polar molecule in an electric-field-free environment is zero. However, in the presence of a weak electric field E (uEkBT), the dipole will be weighted along the field and the time-averaged dipole moment will no longer vanish. Instead will be equal to oriE. The mutual interaction between two polar molecules due to orientation polarizability is identical to the time-averaged dipole-dipole Keesom interaction.

2.1.1.1c. London dispersion forces

These long-range forces, also called electrodynamic forces, induced-dipoleinduced-dipole forces, or charge-fluctuation forces, are quantum mechanical in origin. They act between all atoms or molecules, even totally neutral and non-polar atoms such as helium. To understand this interaction, consider an atom in the Bohr's model, where the electrons are orbiting around the nucleus. For a non-polar atom, the time-averaged dipole is zero. However, the instantaneous position of the electrons leads to a finite dipole moment. This dipole will fluctuate according to the charge distribution in the atom. The electric field arising from such a fluctuating dipole can polarize a nearby molecule or atom, inducing a dipole in it. The interaction between the charge-fluctuating dipole and the induced dipole leads to dispersion forces. The dispersion interaction can be seen as a quantum mechanical polarization interaction.

A summary of some of the above-mentioned interactions is given in Table 21, together with their

respective intermolecular distance r and temperature T dependencies. [14]

8

Type of interaction Common name Interaction energy

Charge Charge Coulomb or ionic interaction 1/r Charge Fixed Dipole 1/r2 Charge Freely rotating dipole 1/(kBTr4) Charge Induced dipole 1/r4

Freely rotating dipole Freely rotating dipole Keesom interaction 1/(kBTr6) Freely rotating dipole Induced dipole Debye interaction 1/r6

Induced dipole Induced dipole London dispersion interaction 1/r6 Table 21: Common interactions between atoms, ions and molecules. The interaction force is obtained by differentiating the energy with respect to the distance r. The Keesom, Debye and London dispersion interactions have the same intermolecular distance dependencies, and together constitute the van der Waals interaction. The attractive covalent, metallic and hydrogen-bonding interactions, as well as the repulsive interactions, are not included in this table.

2.1.1.2. Van der Waals forces

Van der Waals forces were first described by the physicist and Nobel Prize winner Johannes Diderik van der Waals [15] who derived the famous equation of state for liquids and gases: (P + a/V2)(V - b) = RT. Thus, he took into account the effects of the attractive intermolecular forces (the van der Waals forces) by adding the term a/V2 to the pressure P, and the finite size of the molecules by subtracting the term b from the molar volume V. The van der Waals forces are the combination of three distinct forces between freely rotating molecules, such as those found in liquids or gases. These three forces are the orientation force, the induction force and the dispersion force: FvdW = Forient + Find + Fdisp. The three forces correspond to the Keesom interaction, the Debye interaction and the London dispersion interaction, respectively (see Table 21). All three have a 1/r6 dependency in their free energies, where r is the intermolecular distance. Van der Waals interactions between non-polar molecules consist of London dispersion interactions only. Van der Waals interactions with a hard-core repulsion can be described by the Lennard-Jones potential: E(r) = CvdW /r6 + Crep /r12.

Dispersion forces generally exceed the dipole-dependent induction or orientation forces except for small and highly polar molecules, such as water. Furthermore, in an interaction involving to dissimilar molecules of which one is non-polar, the van der Waals interaction is almost completely dominated by the dispersion contribution. The London dispersion interaction feels also from the retardation effect. The retardation effect takes place when the two atoms (or molecules) are separated by a large distance, i.e. the fluctuation period of the instantaneous dipole is comparable with the time needed for the electric field to propagate from one atom to the other. When the electric field from the induced dipole reaches the initial dipole, the direction of the initial dipole has already changed, with the consequence that the attractive dispersion interaction becomes weaker. Thus, for large distances, the dispersion interaction between two atoms decays faster than 1/r6, approaching a 1/r7 dependency. Note that the Keesom and Debye interactions remain non-retarded at all separations.

9

2.1.2. Molecular and atomic condensates

2.1.2.1. Intramolecular interactions (atoms form molecules)

The simplest molecule is composed of two covalently bound hydrogen atoms. The bond between those two atoms emerges from the covalent interaction, which is quantum mechanical in nature. This interaction can be understood as a common sharing of the valence electrons of both H atoms. Quantum mechanically, it is an overlapping of two electronic s-orbitals to form a -bond. More generally, a covalent bond can be described by a linear combination of atomic orbitals (LCAO). The wave function that describes quantum mechanically the interaction of such bonds has not a broad expansion, compared to the interatomic separation. The covalent forces are therefore of short range, and the bonds have a directionality, that is, they are spatially oriented relative to each other. For example, a C or Si atom, with its s2p2 electronic structure, has four valence electrons. Figure 23 shows the s and the px, py, pz orbitals (each p-orbital exhibits two lobes).

s-orbital

p-orbitals

sp-orbitals sp2-orbitalssp3-orbitals

-bond

-bond-bond-bondCC

C

C

Hybridizationsp sp2 sp3

Bonding

Alignment In plane(120)In line(180)

Tetrahedral(109.5)

: -bond: -bond

CCCCCC

CC

Hybridizationsp sp2 sp3

Bonding

Alignment In plane(120)In line(180)

Tetrahedral(109.5)

: -bond: -bond

Figure 23: With four valence electrons, a C or a Si atom has an s2p2 electronic structure. If such an atom is bonded, the s-orbital can hybridize with the p-orbitals to form sp, sp2 or sp3 hybrid orbitals. A covalent bond between two atoms is formed by an overlap of two orbitals (each atom shares a valence electron). In a -bond, the overlap occurs in two places and not directly between the nuclei (usually between p-orbitals), and in a -bond, the overlap region occurs between the nuclei of the atoms. In molecules, C or Si atoms can thus form single, double or triple covalent bonds.

If such an atom is bonded, the s-orbital can hybridize with one, two or all three p-orbitals to form two sp, three sp2 or four sp3 hybrid orbitals respectively. There is an angle of 180 between the two sp orbitals. The three sp2 orbitals lie in a plane perpendicular to the remaining p-orbital. They are separated by 120. The four sp3 orbitals, which form a tetrahedral structure, are separated from each

10

other by 109.5. These orbital orientations influence the directionality of the bonds involving the respective orbitals. Depending on the hybridization, C can build single, double or triple covalent bonds with two, three or four neighboring atoms, with the above defined orientation between each bonds (see Figure 23).

In a covalent bond between two identical atoms, the electron density is symmetrically distributed between the two nuclei. However, two different atoms involved in a covalent bond can possess different electronegativities. The electron density is higher around the most electronegative atom at the expense of the other, which deforms the electronic cloud (Figure 24). As a result, the two atoms are partially and oppositely charged. The bond has therefore an ionic component and is no longer purely covalent. Another important consequence of this difference in electronegativity is that the bond acquires a polar character, and the molecule can have a dipole moment.

H H H Cl

+

Figure 24: The interaction between the atoms of an H2 molecule is purely covalent, with an electron density that is symmetrically distributed. In a covalent bond between two atoms with different electronegativities, the electronic density is no longer equally distributed. The binding interaction between the atoms in a H+Cl molecule exhibits therefore both a covalent and an ionic character. The asymmetry of the electronic cloud makes the bond polar.

2.1.2.2. Intermolecular interactions (molecules form a condensate)

To form a molecular condensate, the molecules must interact attractively. Depending on the nature of the condensate (e.g. involving one or several kinds of molecules) different interactions can occur. One can cite the above-mentioned van der Waals forces and H-bonds. For non-polar molecules, such as benzene or hydrocarbons, the condensate exists solely because of the London dispersion interactions. These condesates are known as van der Waals liquids or solids, and the molecules interact in a weak and undirected manner. In a condensate composed of polar molecules, there are additional attractive intermolecular forces resulting from the dipole-dipole interactions. The dipole can be permanent (Keesom) or induced (Debye). Moreover, the character of a molecule can be influenced by the environment, such as when the molecule is dissolved in a solvent such as water. An amine group of a non-charged molecule exposed to water can be protonated (NH2 becomes NH3+), or an alcohol group can be deprotonated (OH becomes O), so that the molecules have a net charge and their dipole moment is modified. In this way, the interactions involving polar molecules or ions in solvents can become very complex.

2.1.2.3. Interatomic interactions (atoms form a condensate)

An atomic condensate, made of atoms instead of molecules, usually has a crystalline structure in the solid state. The three main types of bonds that form a crystal are the ionic, the covalent and the metallic interaction.

11

The ionic bond is an electrostatic interaction between positively and negatively charged ions, involving usually a halogena), such as in Na+Cl. This interaction can be reasonably well explained by classical theory. In ionically bonded crystals, no free electrons are present, since they are quasi-bonded to the halogen. Therefore, ionic crystals have poor electrical and thermal conductivities.

An example of a covalently bound crystal is the diamond lattice of C or Si atoms in the sp3-hybridized form. The covalent bond in the crystal is identical to the intramolecular covalent bond described above. Due to the high strength of the bonds, such crystals are stiff and brittle. Identically to ionically bound crystals, covalently bound crystals conduct heat and electricity poorly.

The atoms that form metallic crystals have only a few, weakly bound, valence electrons. In contrast to the covalent bond where the shared electrons are confined near the bond, the valence electrons of a metallic crystal have an expanded wave function, compared to the interatomic distance. As a result, the valence electrons are delocalized over the entire crystal. Furthermore, the metallic interaction can only occur when the crystal is composed of a large number of atoms. Metallic crystals are more ductile than covalently bound crystals, and are typically good electrical and thermal conductors.

The forces in a real atomic crystal are best described as a mixture of these three specific interactions.

The intermolecular or interatomic forces leading to the formation of a condensate (whether liquid

or solid) are known as cohesion forces.

a) Elements found in the group VII of the periodic table, such as fluorine, chlorine, bromine, or iodine.

12

2.2. Interactions between macroscopic bodies

2.2.1. Surface Free Energy and Cohesion The surface tension of liquids is a phenomenon that we observe daily. Water striders running on the

surface of a pond without sinking, or rain droplets captured on a spiders web, are examples of surface tension leading to unusual effects.

Surface tension is related to the attractive cohesive forces acting between the molecules (or atoms) of a liquid. In the bulk liquid, one molecule is isotropically surrounded by other molecules, so that the resultant intermolecular force exerted by all the neighboring molecules is zero. However, the resultant intermolecular force acting on a molecule located at the surface of the liquid is non-zero because of the missing neighbors in the upper half-space. This non-vanishing force points inwards into the liquid. Therefore, work is required to bring a molecule from the bulk to the surface, and thus to increase the surface area. This work can be regarded as a surface tension, also described as surface free energy. To reach thermodynamic equilibrium, a one-component system must minimize its surface. For this reason, a drop in a gravity-free environment takes on a spherical shape, since the sphere is the geometrical object with the smallest surface-to-volume ratio.

The surface tension may equally well be thought of as a force per unit length. Consider a soap film stretched over a rectangular wire frame with one sliding side of length l, as shown in Figure 25.

F

x

l

FF

F

x

l

FF

Figure 25: Left: To reversibly stretch a soap film over a wire frame with one sliding side, a force F must be applied to the movable side. The force F is independent of the value of x. Right: A mechanical analogy to surface tension. The force needed to raise a weight can appear as a horizontal tension.

The sliding side can be reversibly moved along the x-axis. To keep the film stretched a force F must be applied to the sliding side. Thus the surface tension is defined by:

lF = 2 (2-1)The factor 2 arises because the soap film has an upper and a lower surface. The surface tension can also be considered as a surface free energy, which is defined as the reversible work per unit area to create a surface. Moving the sliding side over a distance x requires an energy E equal to Fx, while the surface area of the soap film is increased by a value A = 2lx. Therefore, the surface free energy can also be defined as:

AE = (2-2)Equation (2-2) differs from equation (2-1) only by a factor x, and both describe the same physical concept.

13

The strength of the surface tension depends on the strength of the intermolecular interaction. A liquid condensed by dispersion forces only will therefore have a weaker surface tension than a liquid formed by dipole-dipole interaction or H-bonds. Surface tension values for some liquids are given in Table 22. The intermolecular forces are temperature dependent, and thus, the surface tension is also influenced by the temperature. An increase in temperature will lower the surface tension. For water, the surface tension at 0 C is 75.6 mN/m, and drops to 58.9 mN/m at 100 C [16].

Interactions [mN/m]

n-Pentane (C5H12) vdW 16.1 n-Octane (C8H18) vdW 21.6 n-Hexadecane (C16H34) vdW 27.5 Glycerol (C3H8O3) vdW + H-bond 63.4 Water (H2O) vdW + H-bond 72.8 Mercury (Hg) Metallic 486.5

Table 22: Surface tension for different liquids at 20 C [14,17].

As already seen, the intermolecular forces generate the surface tension of a liquid, and are also related to the cohesion energy. The relationship between surface free energy, , and cohesion energy, WC, is shown in the gedanken experiment, in which a cylinder of liquid is split into two parts. The work needed to separate a liquid cylinder of unit area is equal to the cohesion energy (see Figure 26). However, at the same time, the surface area of the liquid is increased by two unit areas. Thus:

CW 21= (2-3)

unit

area W C

Figure 26: To reversibly separate a cylindrical condensate with a unit cross-section area into two parts in vacuum or in its own saturated vapor, a work WC is required, which corresponds to the cohesion energy.

The same physical concepts of surface free energy and cohesion also exist for a solid-phase condensate. For a solid, the process of forming a new fresh surface is divided in two distinct steps. First, the solid is cleaved to expose a new surface, keeping the atoms in the same position as in the bulk. Second, the atoms of the surface region rearrange to their equilibrium position. For liquids the two steps occurs as one, and the thermodynamic equilibrium is almost instantaneously reached because of the high mobility of the molecules. However, for solids that are far below their melting point, the rearrangement of the surface atoms in a freshly cleaved surface occurs slowly.

Moreover, in contrast to liquids where a surface increase always occurs by moving molecules from the bulk to the surface, a solid surface area can also be increased by elastic stretching. Such elastic deformation leads to a surface stress, defined as the reversible work per area to stretch a surface elastically. A change in the surface stress can e.g. be measured by the bending of a cantilever [18,19].

14

For a crystal, the cohesion energy is dependent on the crystallographic plane, and the surface energy shows an identical dependence [20]. Some approximate values for the surface energies of solids are given in Table 23.

Involved interactions [mJ/m2]

Aluminum Metallic 1100 Silver Metallic 1500 Copper Metallic 2000 Iron Metallic 2400 Tungsten Metallic 4400 Silicon Covalent 1400 PTFE (Teflon) vdW 18

Table 23: Surface free energies of some solids at room temperature [17,21]. The values given for the crystalline materials are only approximate, since they depend on the crystallographic plane.

The surface energies for different solids span two orders of magnitude (compare tungsten with PTFE). Strongly bonded solids will keep their high surface energy only if the surface is protected from surface contamination, for example in vacuum or in inert gas environment. When a solid is exposed to an ambient environment, its high surface energy will be reduced by the adsorption of molecules that have a lower surface tension. E.g., a metallic surface exposed to air will be quickly covered by a thin film of water or hydrocarbon molecules that reduce the surface energy. PTFE has an extreme low surface energy. Thus, adsorption on a PTFE surface is usually energetically less favorable. This gives to the PTFE surface its anti-adhesive property.

2.2.2. Interfacial Energy and Adhesion The previous chapter described the surface free energy, , for a one-phase system, the solid or

liquid phase being exposed to a vacuum or to its own saturated vapor. One should now consider the interactions between two different condensed phases.

The adhesion energy, W12, is the reversible work per unit area needed to separate the two phases in a vacuum, as shown in Figure 27. Indices 1 and 2 denote the two different media, which are usually represented by S for solids and L for liquids. If the two media are identical, the adhesion energy becomes the cohesion energy: W11 = WC.

12WMedium 2Medium 1

unit

area

Figure 27: The adhesion energy W12 is the reversible work that is needed to separate medium 1 from medium 2 in a vacuum. Since all media attract each other in a vacuum, the adhesion energy is always positive.

15

Medium 1

Medium 2(a) (b) (c)

Figure 28: Increase of interfacial area.

The increase of an interfacial area proceeds in three steps, as shown in Figure 28. Firstly, the surface of the first medium is increased in vacuum (a), secondly the same is done for the other medium (b), and thirdly the two newly created areas are brought in contact (c). Thus, the interfacial energy 12 (or interfacial tension), which is defined as the work needed to increase the interfacial area by one unit area, is given by:

122112 W += (2-4)

2.2.2.1. Film pressure

If a condensed phase is exposed to a gaseous phase (denoted as G), gas molecules can adsorb on the surface of the condensed phase, thus changing its surface properties including surface energy. The adsorption of gaseous molecules on a surface is associated with a reduction of the surface energy of the condensate. This decrease in energy is the film pressure, , also known as surface pressure or spreading pressure. Let i be the surface energy of a condensate exposed to its own saturated vapor, and iG the interfacial tension of the condensate exposed to another gaseous phase. Then:

LGLLG = LGLLG = (2-5) SGSSG = SGSSG = (2-6)

In an ambient environment, adsorbed water or hydrocarbons can significantly change surface properties. For example, mica cleaved under high vacuum has a surface energy of S 4500 mJ/m2, whereas cleaved in a humid environment, the interfacial energy decreases to SG 300 mJ/m2, due to water adsorption [14]. The film pressure of an adsorbed layer on a liquid phase can be measured by a Wilhelmy plate balance. The Wilhelmy plate will be discussed in section 2.3.5.1.

Figure 29 summarizes the difference between the concepts of surface energy, interfacial energy and film pressure.

(A) (B) (C)

Figure 29: A) A condensed surface (solid or liquid) in equilibrium with its own vapor. Such a surface is characterized by its surface tension, i, also called surface energy. B) The interfacial energy, ij, characterizes the interface between two condensed phases. C) A condensed phase exposed to a gaseous phase of a different composition can adsorb gas molecules on its surface. The interfacial energy, iG, in such a case corresponds to the surface energy, i, minus the film pressure, iG, of the adsorbed layer.

16

Usually, all solids exposed to a gas phase exhibit a positive interfacial energy. For this reason a highly porous material can sinter into a solid bulk at high temperature. Nevertheless, negative surface energies can be found, e.g. on some crystalline structures of alumina [22].

2.2.3. Forces between condensed bodies

2.2.3.1. Adhesion forces between macroscopic objects

The force arising from the adhesion between contacting bodies will now be considered. For two spherical rigid objects in a vacuum, one may use the Derjaguin approximationa) to determine the attractive adhesion force [14]:

1221

2121 2 WRR

RRFSS

SSspheresphere +

= (2-7)

122 WRF surfacesphere = (2-8)The values R, RS1, and RS2 correspond to the spheres' radii, and W12 to the adhesion work per unit area. If the two objects are composed of the same material, the adhesion work, W12, is replaced by the cohesion work, W11 = WC = 2 S. If the objects are placed in a liquid or gaseous environment, S is replaced by SL or SG respectively. Since SL (or SG) is smaller than S, the adhesion force will also be smaller than in a vacuum.

2.2.3.2. Van der Waals forces between macroscopic objects

In section 2.1.1.2, we saw that the van der Waals (vdW) interaction between two molecules or atoms is of the form E(r) = CvdW /r6. To investigate the force generated by the vdW interaction acting between macroscopic objects, one may sum (or integrate) the interaction energies of all atoms in the first object with all atoms in the second object. This can be expressed as:

=1 2

62211

21dd

V VvdWbodybody r

VVCE (2-9)

where 1 and 2 are the number of atoms per unit volume of the bodies, and r the distance between the two infinitesimal volume elements dV1 and dV2. For simple objects such as flat surfaces, cylinders or spheres, the integral can be approximated, under the condition that the spacing s between the objects must be much smaller that the characteristic size of the objects. Some examples are:

21

2121 6 SS

SSspheresphere RR

RRs

AE +=

(2-10)

sRAE surfacesphere 6= (2-11)

a) The Derjaguin approximation gives the force between two spheres in terms of the free energy per unit area of two flat surfaces at the same separation. For this approach to be valid, the radii of the spheres have to be much larger than the separation distance. For two spheres in contact, the free energy becomes the adhesion energy W12.

17

212 sAE surfacesurface = (2-12)

The values R, RS1, and RS2 correspond to the radii of the spheres. One may notice that the interaction energy decays with 1/s or 1/s2 respectively, a much slower decay than the 1/r6 dependence of the intermolecular pair interaction. In equations (2-11) the surface is assumed to be infinitely large. Equation (2-12) corresponds to the energy between a unit area of one surface and an infinite large area of another surface. The constant A appearing in all three equations is called the Hamaker constant, and is defined as:

212 CA vdW = (2-13)

The Hamaker constant, A, contains all the information about the material properties related to vdW interactions, so that the interaction energy between the two bodies becomes a function of A times an expression related to the geometry. Typical values for the Hamaker constant of condensed phases (liquid or solid) in vacuum or air are around 10-19 J [23]. From equations (2-10) to (2-12), one can easily determine the force between the bodies using F = E/s. E.g. for the sphere-surface pair, one obtains the attractive force:

26sRAF surfacesphere= (2-14)

For a sphere of radius R = 100 nm separated from the surface by s = 10 nm, one obtains an attractive force of about 0.1 nN, and for s = 1 nm the force increases to 10 nN.

The difference between equations (2-8) and (2-14) lies in the fact that the first equation describes the adhesion force between contacting objects, whereas the second describes the attractive force due to the vdW interaction between non-contacting objects separated by a distance s.

18

2.3. Liquids on surfaces

2.3.1. Laplace pressure The interfacial tension described in the previous sections can be seen as a two-dimensional pressure

acting in the plane of the interface. The effect of that interfacial tension on a curved liquid-gas interface will be described here. The curved interface between two immiscible liquids can be described in the same manner. Two approaches are possible, a more mechanical one, and a thermodynamical one.

To explain the mechanical effect of the surface tension on the interface, consider an interface that is slightly bend, like a cylindrical surface, as sketched in Figure 210 A. Let the bending radius be R. To keep the interface bent, one has to apply an excess pressure P inside the bent surface. The forces per unit length are in equilibrium when:

( )RPsinRP LGLG == 2d2d (2-15)

If the interface is now curved in two directions, like an ellipsoidal surface, an additional excess pressure must be applied in order to compensate for the bending in the second direction. The equation therefore becomes:

+=

21

11 RR

P LG (2-16)

Equation (2-16) is known as the Young-Laplace equation, and P corresponds to the Laplace pressure.

It is worth considering in detail the two radii R1 and R2, and the associated excess pressure P. The expression (1/R1+1/R2) corresponds mathematically to the mean curvature of a surface, which is the inverse of the mean radius of curvature RK.

21

111 RRR

K

+== (2-17)

For an arbitrary surface, the mean curvature can be different at each point of the surface. To determine R1 and R2 at a point P on the surface, define a normal to the surface through the point P. The intersection of the surface with a plane containing the normal will define a line. Let the curvature of that intersection line at point P be R1. The second radius of curvature is obtained by defining a second plane, containing also the normal, and perpendicular to the first plane. The intersection of this second plane with the surface gives a second curve, with curvature R2 at point P. Even if the choice of the angular orientation of the first plane influences the value R1, the sum of the inverse of the two radii of curvature remains invariant (although the second plane is always perpendicular to the first).

Until now, no attention was paid to signs of radii R1 and R2. Clearly a surface like a sphere will not have the same mean surface curvature as a saddle point. Since the surface represents the liquid-gas interface, one can chose the Laplace pressure to be:

GL PPP = (2-18)

19

with PL the pressure in the liquid phase and PG in the gaseous phase. The sign of R1 or R2 will be defined, accordingly to equation (2-18), by the direction of the center of each radius: positive if the center points into the liquid, and negative if the center points towards the gas phase. Thus, the interface of a liquid droplet will have a positive mean surface curvature, whereas an air bubble imprisoned in a liquid will have a negative mean surface curvature. A saddle point as shown in Figure 210 C has two radii of curvature with opposite signs.

A)

d LGLG

R1 dzR2

d2d1

R

P

R2

R1

Liqu

id

A

ir

C)

B)

R1 > 0 R2 < 0

Figure 210: A and B) Equilibrium conditions for a curved liquid-gas interface. C) The two radii of curvature R1 and R2 that define the mean surface curvature .

Equation (2-16) was derived from a mechanical point of view. The same can be done by considering the surface energy. The interfacial energy LG was defined to be the energy necessary to increase by one unit area the interfacial surface. Thus, to minimize the total surface energy, the liquid-gas interface should have the minimum surface area. Because of the conservation of mass, a reduction in interfacial area is accompanied by a change of pressure in the media enclosed by the interface. Consider an interfacial surface, spanned over two angles d1 and d2 lying in perpendicular planes as shown in Figure 210 B. The area of the initial surface element is defined by the two radii of curvature R1 (in the plane of d1) and R2 (in the plane of d2): A(z) = R1d1R2d2. If the surface is displaced to the right by a distance dz, the new area will be A(z+dz) = (R1+dz)d1(R2+dz)d2, so that in a first order approximation the increase of the surface will be: dA = (R1+R2)d1d2dz. There will be a pressure difference P across the surface, acting over the area A(z). In equilibrium, one has LGdA = PA(z)dz:

( ) zRRPzRRLG dddddd 22112121 =+ (2-19)or:

+=

+=

2121

21 11RR

RRRRP LGLG (2-20)

which is again as expected the Young-Laplace equation. It is important to note, that the radius with the smaller modulus contributes most to the Laplace pressure.

20

The Young-Laplace equation is one of the fundamental relationships in the description of capillarity or other phenomena associated with liquid surfaces. The equation tells us that the crossing of a curved liquid-gas (or liquid-liquid) interface is always accompanied by a change in pressure. The change in pressure is proportional to the mean surface curvature of the interface. A higher interfacial curvature results in a higher Laplace pressure.

It is noteworthy that a two-dimensional pressure (the surface tension) can generate a true three-dimensional pressure.

2.3.2. Contact angle When a drop of liquid is placed on a solid surface, the drop can either spread over the surface, or

remain in a ball-like shape. The driving force to reach the final state is the tendency to reduce the total interfacial energies, viz. the solid-gas (SG), solid liquid (SL), and liquid-gas (LG) interfaces. Thermodynamic equilibrium is achieved when the total surface free energy is minimized, with the additional constraint of mass conservation for the droplet (or volume conservation, since a liquid is almost incompressible). The total surface energy is determined as following

( ) const.++=++= LGLGSGSLSLLGLGSGSGSLSLtot AAAAAW (2-21)where Aij is the surface area of the respective ij-interface. At thermodynamic equilibrium, dWtot must be zero. That means

( ) 0ddd =+= LGLGSGSLSLtot AAW (2-22)or

+=

=const.dd

VSL

LGLGSLSG A

A (2-23)

In a free-standing droplet, the minimization of surface energy leads to a spherical drop shape. It is therefore reasonable to assume that a drop sitting on a homogenous surface has a shape like a spherical cap, as shown in Figure 211. Thus, it is possible to work out the differential of Equation (2-23). The detailed derivation is given in Annex 1, section 8.1.

( )AA

VSL

LG cosdd

.const

==

(2-24)

The variable, , is the contact angle, which is the angle between the plane of the surface and a tangential plane to the liquid surface at the three-phase contact line. The angle that lies inside the liquid is given.

LiquidGas

Solid

Figure 211: The contact angle, , of a three-phase contact line in a sessile droplet is defined as sketched. Depending on the surface wettability, the contact angle lies between 0 and 180.

21

Inserting Equation (2-24) in Equation (2-23) leads to the famous Young's equation:

( ) LGSLSG cos+= (2-25)When no adsorption is present, neither on the solid surface nor on the liquid (film pressure 0), Equation (2-25) becomes S = SL + L cos().

A complete derivation, taking gravity into account and without restriction to a spherical-cap shaped drop, leads to an identical result [24].

Young's equation describes the equilibrium in the solid plane of the three forces resulting from the three interfacial tensions, acting on an infinitesimal section of the three-phase contact line. This is shown on the left-hand side of Figure 212. If the solid surface is curved but still smooth, Equation (2-25) remains valid at each point of the three-phase contact line: the force equilibrium in a plane tangential to the solid surface at the point of interest is considered (illustrated on the right-hand side of Figure 212). Thus, Young's equation is independent of the surface geometry as long as the solid surface is smooth. In thermodynamic equilibrium, the contact angle is the same on a flat surface, inside a capillary, or at any point of a three-phase contact line lying on any irregularly shaped surface.

SGLG

SL

SG

LGSL

LGcos()

Figure 212: In thermodynamic equilibrium, the contact angle is defined by the three interfacial tensions acting on an infinitesimal section of the three-phase contact line. As long as the solid surface is smooth, the contact angle is independent of the geometrical shape of the solid surface.

A surface with a large contact angle is described as hydrophobic, whereas a surface with a low contact angle is described as hydrophilic. The denotation "hydrophobic" or "hydrophilic" refers usually to water. Since the contact angle depends on the liquid surface tension, and on the SL interfacial tension, measurements with different liquids on the same surface will result in different contact angle values.

A solid surface having sharp concave or convex edges is no longer smooth (not continuously differentiable). The edges influence the concept of contact angle described above. Let us first consider a section of the three-phase contact line lying on a concave edge. Several "tangential" planes to the edge top can now be constructed, so that the direction of the liquid-gas interface, which arises from the contact line, is no longer unique. Such a situation is sketched on the left in Figure 213. However, on a convex solid edge, there is no plane that both contains the edge and does not cross the solid body. Thus, a three-phase contact line cannot rest on a convex solid edge, but will lie slightly displaced from it, as shown in the left part of Figure 213.

22

Figure 213: Behavior of a three-phase contact line on a sharp edge. On a concave edge (left hand side), one can observe a pinning of the contact line. On the other hand, the contact line cannot rest on a convex solid edge (right hand side).

Let us consider the in-plane force equilibrium sketched in Figure 212; what happens when SG > SL + LG? The contact angle cannot be smaller than zero degree. Young's equation seems also to fail when SL > SG + LG. Let us first consider the latter case, where the SL interfacial energy is higher than the sum of the SG and LG interfacial energies. For a weightless drop lying on a surface, the three-phase system would always gain energy during the separation of the liquid phase from the solid phase. Thus, a complete dewetting will occur, so that there is no contact between the liquid and the solid surface.

2.3.2.1. Spreading coefficient

When the SG interfacial energy exceeds the sum of the SL and LG interfacial energies, an increase in the SL interface area, together with an increase of the LG surface, and at the cost of SG surface, will reduce the total interfacial energy. Thus, the liquid will spontaneously spread over the solid surface to form a thin film. The contact angle can clearly not be used to characterize this effect. Instead, the spreading coefficient S is used, defined in the following way:

( )LGSLSGi SG/LG S += (2-26)The index "i" refers to the fact, that initially no molecules of the liquid are adsorbed at the solid-gas interface. A spreading coefficient is positive if the spreading is accompanied by a decrease in free energy, that is, when it occurs spontaneously.

The spreading that occurs when a liquid with a low surface tension is placed on a liquid with a high surface tension, such as toluene on water, can be described in a similar manner. However, when two liquids are in contact, diffusion occurs until they become mutually saturated, which also influences their interfacial tensions. For a benzene drop placed on a water surface, the benzene lens first rapidly spreads out, and then, as mutual saturation occurs, the benzene retracts back to a lens. The water that is left behind is not pure, but is saturated and recovered by a thin layer of benzene. For pure water at 20 C, the surface tension is 72.8 mN/m. Nevertheless, the film pressure (see equation (2-5)) due to the thin benzene layer, lowers the surface tension to 62.2 mN/m, leading to a final spreading coefficient S f = S i that is negative, inducing the retraction of the liquid back to a lens [17,25,26]. Figure 214 shows the different possible behaviors when a drop is deposited on a substrate.

23

S i < 0 S i > 0 S i > 0 S i > 0S f > 0 S f < 0

A) B) C) D)

Figure 214: Behavior of a liquid drop onto a substrate, solid or liquid. A) With a negative initial spreading coefficient, the drop keeps a lens-like shape. B) With a positive spreading coefficient, immediately after the deposition the drop spreads over the substrate. C) As long as the final spreading coefficient doesn't become negative, the deposited liquid remains on the surface as a thick film. D) When the adsorption of molecules on the substrate leads to a negative final spreading coefficient, only an adsorbed thin film remains on the substrate, while the excess liquid retracts to a lens.

2.3.2.2. Influence of the surface roughness on the contact angle

Young's equation is valid for a smooth and homogenous surface, where the local contact angle, , is defined with respect to a plane that is tangential to the surface at one local point of the three-phase contact line. Other approaches are needed to describe the global (or apparent) contact angle of a droplet on a rough or inhomogeneous surface.

2.3.2.2a. Wenzel's approach If a liquid drop lies on a rough surface, the apparent contact angle ' (measured with respect to the

averaged surface) can be different from the local contact angle (with respect to the tangential plane at one local point of the three-phase contact line). One explanation of this behavior was given by Wenzel [27,28]. He stated that the interfacial energy per unit area is higher on a rough surface than on a flat surface. Thus, Young's equation must be corrected by a factor, r, which is the ratio of the actual area to the projected area of the solid surface (r 1), leading to the Wenzel's equation:

( )'rr LGSLSG cos+= (2-27)or:

( )( ) ( )( )

( )( )

( )

==

=

1cos1cos1

1cos if

1coscoscos

1cos

rr

r

'r'

' (2-28)

Surface roughness will therefore make a hydrophilic surface even more hydrophilic, and on a hydrophobic surface even more hydrophobic. The effect of the roughness will be even more pronounced if the initial surface is highly hydrophilic or highly hydrophobic.

2.3.2.2b. Cassie's approach In the presence of a flat, but inhomogeneous surface, Cassie and Baxter [29,30] developed another

method. This is to assume that the surface is made of two distinct surface properties, with local contact angles, , and 1 respectively. Let f be the fractional part of the first kind of surface, and (1-f) that of the second. The apparent contact angle, ', on the composite surface will therefore be:

( ) ( ) ( ) ( )1cos1coscos ff' += (2-29)For a homogeneous hydrophobic and highly rough surface, air can be trapped among the hydrophobic structures under the liquid, as shown at the right hand side of Figure 215. In a gedanken experiment

24

one can substitute the areas where the liquid doesn't touch the surface with a flat surface with a contact angle of 1 = 180. Therefore, for a hydrophobic and highly rough surface when the SL interface is not fully wetted, one has:

( ) ( ) 1coscos += ff' (2-30)Here, f is the fraction of the area that is wetted by the liquid, and the local contact angle.

In summary, a liquid drop on a hydrophobic and rough surface can be described by the Wenzel's equation (2-28) when the liquid completely wets the solid surface. If air is trapped at the solid-liquid interface, Cassie's equation (2-30) can be used. Both approaches are sketched in Figure 215. It is to date not clear, when one or the other model should be used, since both can exhibit local minima in energy [31].

Wenzel's approach Cassie's approach

Figure 215: Two models describing hydrophobic contact angles on rough surfaces. Wenzel's (wetted contact) and Cassie's (composite contact) theories present two possible equilibrium drop shapes on a rough substrate.

The combination of a high roughness with hydrophobic contact angles can be used to produce so-called superhydrophobic surfaces with apparent contact angles higher than 170. Substrates structured to have an array of small pillars, created by microfabrication techniques [32,33], or substrates with a fractal-like topography [34,35] could be used as surfaces having a high roughness.

2.3.2.3. Contact line tension

Youngs equation (2-25) was derived by minimizing the free energy of the three interfaces. In the same way than molecules inside the liquid have a lower energy that molecules located at the surface, a molecule located at the three-phase contact line can have a different energy than one lying at the interface. This leads to line tension, a one-dimensional analog of surface tension. The contact line tension, , can be given in N or in J/m. Taking account of the contact line tension in equation (2-21) gives a total free energy of:

( ) const.+++= PAAW SLGLGLGSGSLSLtot (2-31)where PSLG is the length of the three-phase contact line. To obtain the thermodynamic equilibrium, Wtot must be minimized. Thus

( ) 0dddd =++= PAAW SLGLGLGSGSLSLtot (2-32)or, together with the boundary condition of volume conservation for the liquid:

+

+=

== const.const. dd

dd

VSL

SLG

VSL

LGLGSLSG A

PAA (2-33)

25

For a spherical-cap shaped droplet, the first differential was already calculated in Annex 1, and is equal to cos(). The second differential can be easily evaluated, using r as the radius of curvature of the three-phase contact line:

rr

r

rr

rAr

P

AP

V

SL

V

SLG

VSL

SLG 1

dd

2dd

ddd

d

dd

2

const.

const.

const.

====

==

(2-34)

Thus, the modified Youngs equation is obtained:

( )r LGSLSG ++= cos (2-35)

which can also be written as

( ) ( )LGr

= coscos (2-36)

where is the contact angle in the limit of a very large drop (r ). Equation (2-36) shows that the contact line tension will only affect the contact angle for a high curvature of the three-phase contact line.

Both the sign and magnitude of the line tension are still under debate [36]. Theoretical calculations predict to be on the order of some 10-11 J/m [36,37], whereas experimental values are much more controversial [36,38,39]. However, assuming the theoretical values are correct, the influence of contact line tension should be experimentally significant only if r < /LG 10-11 Jm-1 / 10-2 Jm-2 = 1 nm. In other words, a measurable influence of the contact line tension is to be expected in the nanometer range, while in large systems, interfacial energies will dominate.

2.3.2.4. Contact angle hysteresis

The derivation of the Young's equation was based on thermodynamics, so that one could imagine a unique and well defined contact angle for a given solid-liquid-gas set. Nevertheless, in experimental observations (see Section 3.2), one usually observes a difference between the contact angle of an advancing three-phase contact line (the line is moved towards the solid-gas interface) and a receding line (moved towards the solid-liquid interface). The advancing contact angle adv is always higher than the receding contact angle rec, leading to the contact angle hysteresis.

It is due to contact angle hysteresis that a drop doesnt roll or flow under gravity on a tilted surface. A low hysteresis will immobilize the drop only on a slightly tilted surface. The immobilization of a drop on a vertical surface can be guaranteed only with a higher hysteresis.

A complete understanding of the hysteresis observed in the contact angle measurements has not yet been achieved. One explanation for the hysteresis can be found in surface heterogeneities and surface roughness.

A heterogeneous surface is shown in Figure 216, where the surface has locally different contact angles. As the drop moves, the advancing three-phase contact line will remain pinned at the hydrophilic-hydrophobic transition, and will start to move only when it is energetically more favorable, i.e. when the local contact angle of drop is slightly larger than the contact angle of the hydrophobic area. When the advancing front leaves the hydrophobic area and enters a hydrophilic one, the three interfacial forces will be suddenly unbalanced, and the advancing contact line will "jump" over the hydrophilic area until a new local minimum in energy is found, e.g. at the next hydrophilic-

26

hydrophobic transition. The contact angle of the advancing contact line will therefore be close to the contact angle of the hydrophobic area. Similarly, the receding contact line will jump from one hydrophilic area to the other. This so-called pinning effect results in contact angle hysteresis.

adv <

rec

Figure 216: The advancing and receding contact angles of a moving drop are influenced by surface heterogeneities (on the orange surface, is larger than on the yellow).

Similarly, on a rough surface, the advancing and receding lines will be pinned on a downhill-slope, arising from a difference between the apparent advancing contact angle adv and apparent receding contact angle rec, even if the local contact angle is identical at both sections of the moving three-phase contact line (see Figure 217).

recadv

recadv

Figure 217: The apparent advancing and receding contact angles are influenced by the surface roughness.

Other reasons that explain the presence of the hysteresis can be the adsorption on the solid surface of molecules dissolved in the liquid, or rearrangement or alteration of the solid surface by the liquid.

The static contact angle sta refers